MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sban Structured version   Unicode version

Theorem sban 2139
Description: Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sban  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )

Proof of Theorem sban
StepHypRef Expression
1 sbn 2130 . . 3  |-  ( [ y  /  x ]  -.  ( ph  ->  -.  ps )  <->  -.  [ y  /  x ] ( ph  ->  -.  ps ) )
2 sbim 2135 . . . 4  |-  ( [ y  /  x ]
( ph  ->  -.  ps ) 
<->  ( [ y  /  x ] ph  ->  [ y  /  x ]  -.  ps ) )
3 sbn 2130 . . . . 5  |-  ( [ y  /  x ]  -.  ps  <->  -.  [ y  /  x ] ps )
43imbi2i 304 . . . 4  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ]  -.  ps ) 
<->  ( [ y  /  x ] ph  ->  -.  [ y  /  x ] ps ) )
52, 4bitri 241 . . 3  |-  ( [ y  /  x ]
( ph  ->  -.  ps ) 
<->  ( [ y  /  x ] ph  ->  -.  [ y  /  x ] ps ) )
61, 5xchbinx 302 . 2  |-  ( [ y  /  x ]  -.  ( ph  ->  -.  ps )  <->  -.  ( [
y  /  x ] ph  ->  -.  [ y  /  x ] ps )
)
7 df-an 361 . . 3  |-  ( (
ph  /\  ps )  <->  -.  ( ph  ->  -.  ps ) )
87sbbii 1665 . 2  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  [ y  /  x ]  -.  ( ph  ->  -.  ps ) )
9 df-an 361 . 2  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) 
<->  -.  ( [ y  /  x ] ph  ->  -.  [ y  /  x ] ps ) )
106, 8, 93bitr4i 269 1  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   [wsb 1658
This theorem is referenced by:  sb3an  2140  sbbi  2141  sbabel  2598  cbvreu  2930  sbcan  3203  sbcang  3204  rmo3  3248  inab  3609  difab  3610  exss  4426  inopab  5005  mo5f  23972  rmo3f  23982  iuninc  24011  suppss2f  24049  fmptdF  24069  disjdsct  24090  esumpfinvalf  24466  measiuns  24571  ballotlemodife  24755  sb5ALT  28609  2uasbanh  28648  2uasbanhVD  29023  sb5ALTVD  29025
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659
  Copyright terms: Public domain W3C validator